The literature of Logic recognizes nonstandard logics, of which modal logic is a prototype.The literature of Mathematics recognizes nonstandard real numbers, but little else.
Why? Because the literature and textbooks and lectures are usually deficient in explaining "The Trinity: Relations, Functions, Operations". Within this context, the reason for the nonstandard number system of my title (and of my creation) readily emerges.
RELATIONS The protypical relation is the binary relation, as shown below.
Definition: B º [1st, 2nd] is a binary relation, B, on a set, S, if, and only if, terms 1st,2nd denote any two members of S, where [1st, 2nd] denotes an ordered set, so that [1st, 2nd] does not necessarily imply [2nd, 1st]. A relation can also exist between relata from different sets, such as marriage between two different genders.
Note that this definition allows, for aRb, existence of both the binary relations, [a, b] and [b, a], equivalently, aRb = bRa, or R(a,b) = R(b,a). Example, the relation of neighbor.
Given a ternary relation of the form R(a,b,c), this can be compactified as [[a, b], c]], a "binary or a binary". Thus, mathematical literature sometimes displays the addition, 2 + 3 = 5 as [[2, 3], 5]].
Clearly, although tedious to write, any n-ary relation, for n > 2, can be written as a binary relation, making this form protypical of finite relations. This is convenient, since it projects onto the typical functions and operations, as we see below
FUNCTIONS We obtain a function from a relation, by invoking a restriction upon it.
Definition: A (binary) function is a (binary) relation such that a given first member relates to at most one second member.
Thus, specification of [a, b] precludes [b, a] in "the same universe of discourse".
Note, however, that [a, b] could allow [c, b], for distinct a,b,c, so that two different first members relate to the same second member. In such a case, the function is multivalent. But this is not the case with a univalent function.
OPERATIONS Before proceeding, I must clarify a confusion of language -- one among many! -- in mathematical literature. The operation of addition is customarily referred to as "a binary relation", although we saw above that it is a ternary relation or binary of binary. For example, 2 + 3 = 5 as [[2, 3], 5]]. A simpler relation is a "unary operation", although, strictly, it fits the binary pattern. (A nullary operation is a constant.)
To proceed, I must take a "unary operation" (binary-patterned) operation such as successor, S(n) º n+1, which can also be written as [n, n+1] .
We noted above that a relation can involve relata from different sets. And we now need to introduce names for these sets. Given [p, q], we label the first-member set as domain set, second-member set as codomain set or range.
Definition: A (binary) function is a (binary) operation if, and only if, its codomain is a subset of its domain.
Thus, successor is an operation on the set of natural numbers since "it sends a natural number into a natural number", that is, any set of successors is a subset of natural numbers.
However, "the catalog function mapping a number in a catalog to an item on a warehouse shelf" is not an operation since the item-set is not a subset of any number-set.
INVERSES OF PRIMARY ARITHMETIC OPERATIONS Elsewhere, I explain the need for, and generation of, subtraction-as-additive-inverse, division-as-multiplicative-inverse. And I show that inversion of an operation is possible if, and only if, the operation is well-defined ("cancellative").
Definition: Operation o is right-well-defined if, and only if, h o j = k o j Þ h = k; left-well-defined if, and only if, h o j = h o l Þ j = l; well-defined if, and only if, both left- and right-well-defined.
Clearly, successor is right-well-defined since [n, n+1] = [n, m+1] Þ n = m; left-well-defined since [n, n+1] = [j, n+1] Þ n = j; hence, it is well-defined. In that "elsewhere", I demonstrate well-definedness of addition, multiplication, providing for inversion. It is easy to show that, implicitly, this requires that the operation be a univalent function.
In primary or middle school, you learned operations from multivalent non-well-defined operations: least-common-multiple (LCM), greatest-common-factor (GCF). Thus, LCM(3,2) = LCM(3,6) = 6 but, 2 ¹ 6, so LCM is not right-well-defined, which suffices to show non-well-definedness and noninversivity of LCM. Similarly, GCF(6,10) = GCF(14,22) = 2, clearly displaying failure of both left- and right-well-definedness, hence, well-definedness and inversivity.
Thus, LCM(7, 2) = LCM(7, x) = 14 has an INFINITE SET OF SOLUTIONS: THE POWERS OF 2. Perhaps others. Again, GCD(42,66) = GCD(42,x) = 42 has AN INFINITE SET OF SOLUTIONS.
Why? Because, although LCM, GCD are operations, both are also bmultivalent functions (whereas addition, multiplication are univalent functions). This is critical because -- as shown in the literature -- these operations are, respectively, homologous to disjunction, conjunction in statement logic; to join, meet in lattices; to the primary operations in "boolean algebra" and in logic circuitry, etc..
But there is no follow-up on this. For example, because multiplication and division are univalent operations, it is possible to state the following.
Fundamental Theorem of Arithmetic (FTA): A number in rational number arithmetic can be written as the product of prime factors in exactly one way (if order of factors is ignored).
Since integers and naturals are "subsystems" of the rationals, this follows for them. But a supposed proof for "Fermat's Last Theorem" foundered when it was pointed out that FTA fails for complex numbers.
This much is well known and documented in the literare. But it is ignored that the equivalence of FTA fails for the multivalent operations of LCM and GCF in arithmetic.
This "opens the door" to freenats, frintegers.
But before I "drag you through that" (if, indeed, you stick with it), you've the right to query: "So what?"
Briefly, two subjects depend upon this, probabilities and data-bases. (I've dealt with both because I've taught many statistics courses and contributed to the field, and I built science data-bases for Naval Reseach Laboratory, using a beta-version of relational data-base theory which was extrapolated into the giant firm of Oracle.) And, in each case, the current "universe of discourse" being considered is "penny-ante" compared to what it could be by drawing upon freenats and frintegers.
The probabilities affect your health, security, political freedom, etc. And, whether or not you are an investor, your pension fund certain depends on investing, which depends on data-base information.
So, your life, liberty, and pursuit of happiness, as well as your pocket-book, depend upon this. Nuff said? And if you take up my work and develop it, that could pay off big!
The language above is in terms of integers, which is typical of the literature. But it could as well have been rendered in terms of natural numbers.
Now look at freenats and frintegers.